Exercise 17, § 4

Let G be a compact group and H a closed subgroup of G. Let $\varrho$ be a unitary representation of H in a finite-dimensional Hilbert space E. a) Let $\pi \in \widehat{G}$. The space Hom$_G(\pi$, Ind$^G_H(\varrho ))$ is of finite dimension.

b) For every $u\in$ Hom$_G(\pi$, Ind$^G_H(\varrho ))$, the image of $u$ is contained in the image of $\mathscr{C}(G; E)$ in $L^2(G; E)$.

c) There exists a linear mapping Φ from the space Hom$_G(\pi$, Ind$^G_H(\varrho ))$ into the space Hom$_H$(Res$^G_H(\pi ), \varrho$ ) such that, for every $u\in$ Hom$_G(\pi$, Ind$^G_H(\varrho ))$ and every $x\in E_{\pi}, \Phi (u)(x)$ is the value at $e$ of the continuous function representing $u(x)$ d) The linear mapping Φ is an isomorphism.

e) Suppose that $\varrho$ is irreducible. The multiplicity of $\pi$ in Ind$^G_H(\varrho )$ is equal to the multiplicity of $\varrho$ in Res$^G_H(\pi )$ ("frobenius reciprocity formula", cf. A, VIII, p. 392).